We consider a general queueing model with price-sensitive customers in which the service provider seeks to balance two objectives, maximizing the average revenue rate and minimizing the average queue length. Customers arrive according to a Poisson process, observe an offered price, and decide to join the queue if their valuation exceeds the price. The queue is operated first-in first-out, and the service times are exponential. Our model represents applications in areas like make-to-order manufacturing, cloud computing, and food delivery. The optimal solution for our model is dynamic; the price changes as the state of the system changes. However, such dynamic pricing policies may be undesirable for a variety of reasons. In this work, we provide performance guarantees for a simple and natural class of static pricing policies which charge a fixed price up to a certain occupancy threshold and then allow no more customers into the system. We provide a series of results showing that such static policies can simultaneously guarantee a constant fraction of the optimal revenue with at most a constant factor increase in expected queue length. For instance, our policy for the M/M/1 setting allows bi-criteria approximations of $(0.5, 1), (0.66, 1.16), (0.75, 1.54)$ and $(0.8, 2)$ for the revenue and queue length, respectively. We also provide guarantees for settings with multiple customer classes and multiple servers, as well as the expected sojourn time objective.

翻译：本文考虑一个具有价格敏感型顾客的通用排队模型，服务提供商需平衡两大目标：最大化平均收入率和最小化平均队列长度。顾客按泊松过程到达，观察报价后若其估值高于价格则决定加入队列。队列采用先到先服务模式，服务时间服从指数分布。该模型适用于按单制造、云计算及食品配送等领域。此模型的最优解是动态的——价格随系统状态变化而变化，但此类动态定价策略可能因多种原因不可取。本研究针对一类简单自然的静态定价策略（在达到特定占用阈值前收取固定价格，此后禁止新顾客进入系统）提供性能保证。我们通过系列成果证明，此类静态策略能同时保证最优收入的恒定比例，且预期队列长度的增幅不超过恒定倍数。例如，针对M/M/1场景的策略可分别实现收入与队列长度的双准则近似值$(0.5,1)、(0.66,1.16)、(0.75,1.54)$和$(0.8,2)$。此外，我们还为多顾客类别、多服务器场景以及预期逗留时间目标提供了性能保证。